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A Loophole in Bell’s Theorem?
Parameter Dependence in the Hess-Philipp Model
Wayne C. Myrvold1
Department of Philosophy
University of Western Ontario
London, ON Canada N6A 3K7
[email protected]
ABSTRACT
The hidden-variables model constructed by Karl Hess and Walter Philipp is
claimed by its authors to exploit a “loophole” in Bell’s theorem; according to
Hess and Philipp, the parameters employed in their model extend beyond those
considered by Bell. Furthermore, they claim that their model satisfies Einstein
locality and is free of any “suspicion of spooky action at a distance.” Both of
these claims are false; the Hess-Philipp model achieves agreement with the
quantum-mechanical predictions, not by circumventing Bell’s theorem, but via
Parameter Dependence.
KEYWORDS: Bell's Theorem, Nonlocality, Hess, Philipp, Parameter Dependence.
1. Introduction. In recent papers, Karl Hess and Walter Philipp (2001a,c) exhibit a hiddenvariables model that reproduces the quantum-mechanical statistics for the EPR-Bohm
experiment. Of this model, they make the following claims:
1. The parameters used in the model go beyond those considered in the proofs of Bell’s
theorem, rendering the theorem inapplicable to their model.
2. The model satisfies the condition of Einstein locality and is free of “spooky action at
a distance.”
Both of these claims are false, as will be shown below.
2. Factorization, Outcome Independence, Parameter Independence. This section will
introduce terminology and notation that will be used in subsequent sections. Much, if not all, of
the material in this section will be familiar to many readers.
The EPR-Bohm experiment involves a pair of spin-½ particles prepared in the singlet
state. Measurements are performed on each of the pair, at stations S1 and S2, respectively, of spin
in directions a and b, respectively, where a and b are unit vectors. Denote by Aa, Bb the
outcomes, represented by values ±1, of the measurements on particle 1 and 2, respectively. The
quantum-mechanical prediction for the expectation value of their product AaBb is
E(a, b) = –a⋅b.
Assume that the outcome of the experiment depends on a set of parameters denoted by λ.
Without further assumptions about the nature of this dependence little can be said. The crucial
assumption underlying the derivation of Bell’s theorem is an assumption of factorizability of the
underlying probability measures. Let pab(xa, yb| λ) be the probability that, in a state described by
λ, the measurement of spin-a and spin-b on particles 1 and 2, respectively, have outcomes xa, yb,
2
(the variables xa, yb take on values in {-1, 1}). The Bell factorizability condition is the condition
that there exist probability functions pa(xa| λ), independent of b, and pb(yb| λ), independent of a,
such that
pab(xa, yb | λ) = pa(xa | λ) pb(yb | λ).
By constructing an inequality that must be satisfied by any factorizable theory but which
is violated by the quantum-mechanical predictions, Bell was able to prove that no theory
satisfying the Bell factorizability condition can reproduce the statistical predictions of quantum
mechanics for the EPR-Bohm experiment. Bell’s theorem is sometimes glossed as a proof of the
impossibility of a hidden-variables theory, and this is how Hess and Philipp present it: “The
work of Bell [(1964)] attempts to show that a mathematical description of EPR-type experiments
by a statistical (hidden) parameter theory is not possible” (2001c, 14228).
If this is what Bell
had attempted to do, then his efforts would have been in vain—as Bell himself pointed out. In
the very paper cited, Bell exhibits a simple hidden-variables model for the EPR-Bohm
experiment, a model which is, of course, nonlocal. In his work, Bell frequently emphasized that
it is not possible to show tout court that no hidden-variables theory is possible, as we actually
have such a theory in Bohm’s theory (Bohm 1952a,b). The paper (1966) that preceded (in order
of composition, but not publication) the “Bell’s Theorem” paper (1964) is devoted to criticizing
purported no-go theorems; this includes the theorem that has come to be known as the KochenSpecker theorem (Kochen and Specker, 1967), and which was proved, independently of Kochen
and Specker, by Bell in his paper. Bohm’s theory served Bell well as a counterexample to
purported no-go theorems; a theorem purporting to show the impossibility of a hidden-variables
theory must make some assumption not satisfied by the Bohm theory.
In particular, the
contextuality exhibited by the Bohm theory highlights the assumption of noncontextuality
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necessary for the Bell-Kochen-Specker theorem, and the nonlocality exhibited by the Bohm
theory suggested the locality condition used to derive the Bell inequalities.
Jarrett (1984) showed that the Bell factorizability condition can be expressed as a
conjunction of two conditions, the condition called “Locality” by Jarrett (1984; 1989) and
“Parameter Independence” by Shimony (1986), on the one hand, and “Completeness” (Jarrett)
or “Outcome Independence” (Shimony), on the other.
Given the probability functions
pab(xa,yb|λ
λ), we define marginal probability functions:
p ab ( x a | λ ) =
∑p
ab
( xa , yb | λ )
y b ∈{ −1, 1}
p ab ( y b | λ ) =
∑p
ab
( xa , y b | λ )
xa ∈{ −1, 1}
Parameter Independence is the condition that pab(xa|λ
λ) not depend on b and pab(yb|λ
λ) not depend
on a. Outcome Independence is the condition that, for given a, b, λ, the distribution of xa is
independent of yb, and vice versa. That is,
pab(xa, yb| λ) = pab(xa|λ
λ) pab(yb|λ
λ).
A violation of Parameter Independence entails that, for a given value of λ, the statistical
distribution of xa can be changed by changing the setting b of the distant analyzer. If the
parameters λ are observable, and if one can use them to adjust the settings of an analyzer,
Parameter Dependence can be exploited to send a superluminal signal from one station to the
other by manipulating the settings. If, for some reason, the parameters λ cannot be used to adjust
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the settings of the analyzers, no superluminal signals can be sent but the nonlocality of the model
remains, and, since the probability distribution of xa depends on the instantaneous value of the
setting at the other station, a model exhibiting Parameter Dependence requires a distinguished
relation of distant simultaneity and cannot, at a fundamental level, be a relativistic model.
If we take, as the parameter λ, the quantum-mechanical state without any supplementary
variables, the quantum-mechanical probabilities satisfy Parameter Independence but violate
Outcome Independence. The peculiar sort of violation of Outcome Independence exhibited by
quantum mechanics is, at the very least, not obviously at odds with relativity, as a violation of
Parameter Independence would be (for discussion of this issue, see, e.g., Shimony 1978, 1984,
1986; Jarrett 1984, 1989; Ballentine and Jarrett 1987; Butterfield 1989; Teller 1989; Redhead
1986, 1989; Myrvold 2002a, b).
3. An Extended Parameter space? According to Hess and Philipp, the basis for constructing a
hidden-variables model that reproduces the quantum-mechanical statistical predictions lies in
introducing time-dependent correlated parameters associated with the measurement instruments.
Their construction proceeds via the introduction of setting-dependent subspace product measures
(SDSPMs). Let our parameter space Ω be partitioned into disjoint subspaces Ωm. An SDSPM is a
measure, defined on some subspace Ωm, whose dependence on the settings a, b is such that it can
be written as a product of a measure dependent only on a and one dependent only on b. The
probability measure constructed by Hess and Philipp is a sum of setting-dependent subspace
product measures,
µab =
1
N
N
∑ (µ
m =1
5
a
× µb ) m .
Which of the product measures (µ a×µ b)m is in effect on a given run of the experiment will be
determined by the time-dependent correlated parameters.
Hess and Philipp assert that such measures go beyond those in the scope of Bell’s
theorem.
We would like to emphasize that Bell has introduced a number of assumptions on
time dependencies . . . He also introduced a significant asymmetry in describing
the spin properties of the particles and the properties of the measuring equipment.
The spins are described by arbitrarily large sets Λ of parameters. On the other
hand the measurement apparatus is described by a vector of Euclidean space (the
settings), true to Bohr’s postulate that the measurement must be classical. Yet,
the measurement apparatus must itself in some form contain particles with spins
that then, if one want to be self consistent, also need to be described by large sets
of parameters that are related to the settings a, b, c . . . (2001c, 14229).
Now, the original proof of Bell’s Theorem (Bell 1964) did not explicitly take into
account hidden variables in the instruments. In his restatement of the proof (Bell 1971), Bell did
explicitly include such a consideration.2
The instruments themselves could contain hidden variables which could influence
the results. If we average first over these instrument variables, we obtain the
representation
P(aˆ , bˆ) = ∫ dλ ρ (λ ) A (aˆ , λ ) B (bˆ, λ )
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where the averages A and B will be independent of b̂ and â , respectively, if the
corresponding distributions of instrument variables are independent of b and a,
respectively, although of course they may depend on â and b̂ , respectively
(1971, 178–79; 1987, 36–37).
The proof goes through under the assumption that the distribution of the instrument variables
pertaining to one station is independent of the setting of the other (if this assumption is not made,
it is of course possible to reproduce the statistical predictions of quantum mechanics and thereby
violate the Bell Inequalities). The second of Hess and Philipp’s claims, quoted above, is false;
the derivation of the Bell inequalities in no way depends on an assumption that the states of the
measurement instruments are completely described by their settings. This consideration also
applies to the first above-quoted claim, that Bell introduces an assumption that the distributions
of the hidden variables are time-independent; to generate time-dependent distributions, all one
need do is include among the variables a parameter that changes with time in such a way that it
can be used as a clock parameter. The key point is that, if correlations between events at the two
wings of the experiment are due solely to clock parameters in the two instruments, then the
probability of a result at one wing, conditional on a full specification of parameters, including
clock parameters, at that wing, will be independent of the parameters at the other wing, even if
the parameters pertaining to the two stations are correlated with each other.
Although Bell does not exclude the possibility of the measurement results depending on
instrument parameters, he introduces a restriction on such parameters, namely, that the statistical
distributions of parameters of one instrument be independent of the setting of the other
instrument. It should be stressed that the locality condition assumed by Bell will be satisfied by
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clock parameters that are synchronized via some interaction in the common past of the
instruments and subsequently evolve deterministically. If Hess and Philipp’s construction is to
exploit instrument parameters to yield the quantum-mechanical statistical predictions, and
thereby violate the Bell inequality, it must have the probability distribution of the parameters for
one instrument depend on the setting of the other—and it does. This feature of the Hess-Philipp
model has been pointed out by Gill et al. (2002, 5).3 In the next section a simplified version of
Hess and Philipp’s model will be exhibited, which will make the nature of the Parameter
Dependence of such models clear.
4. Parameter Dependence and Signalling. Hess and Philipp claim that their model “is free of
the suspicion of spooky action at a distance.”
This is accomplished by letting the probability measure be a superposition of
setting-dependent subspace product measures with two important properties: (i)
the factors of the product measure depend only on parameters of the station that
they describe, and (ii) the joint density of the pairs of setting-dependent
parameters in the two stations is uniform (2001c, 14232).
That is: the joint distribution of u and v is the uniform measure over a square in —2, and takes the
form,
µab =
1
N
N
∑ (µ
m =1
8
a
×µb ) m .
The index m, which determines, for a given run of the experiment, which subspace measure will
be in play, is a function of the source parameter λ and the time parameters pertaining to the two
stations. For each m, there are functions Am(a, u), Bm(b,v) that determine the outcome of the
experiments at 1 and 2, respectively.
Since µab is the same distribution for every a, b, changing the setting b will not affect the
distribution, on µab, of u, and changing the setting a will not affect the distribution of v.
This
does not, however, guarantee the absence of Parameter Dependence in the model.
Hess and Philipp’s model is rather intricate, and there is not space to give the full details
of the model here. Much of the intricacy, however, stems from the fact that they are concerned
with reproducing the quantum-mechanical statistics for arbitrary settings a, b of the two
analyzers. If we restrict the experimental set-up under consideration to one in which a choice is
made between only two settings for each analyzer—a set-up which, of course, suffices for the
violation of Bell inequalities— it is possible to construct a simpler model that nevertheless shares
the salient features of Hess and Philipp’s model, and, in particular, shares the features that are
touted by Hess and Philipp as being responsible for the spooklessness of their model.
Let a ∈ {0, 1} be a parameter indicating which of the two possible settings has been
chosen for analyzer 1, and let b ∈ {0, 1} indicate the setting of analyzer 2. Let u and v be hidden
variables associated with analyzers 1 and 2, respectively, taking values in [0, 4). Define the
functions σ ai (u ) , τ bi (v) , with values as given in Table 1.
9
1
a
σ (u )
2
a
σ (u )
3
a
σ (u )
σ a4 (u )
u∈[0,1)
a
1–a
a
1–a
u∈[1,2)
1–a
a
1–a
a
u∈[2,3)
a
1–a
a
1–a
u∈[3,4)
1–a
a
1–a
a
1
b
2
b
τ (v )
τ (v )
τ b3 (v)
τ b4 (v)
v∈[0,1)
b
b
1–b
1–b
v∈[1,2)
b
1–b
1–b
b
v∈[2,3)
1–b
1–b
b
b
v∈[3,4)
1–b
b
b
1–b
Table 1
E1
E2
E3
E4
Figure 1
Let Ej, j = 1,...4, be the unshaded regions of the square Ω = [0, 4) × [0, 4) depicted in Figure 1,
and let κj(u,v) be the function equal to 1 if 〈u,v〉 ∈ Ej, and zero otherwise. Define
ij
ρ ab
(u , v) = σ ai (u ) τ bi (v) κ j (u, v) .
It is a simple matter to check that, for every value of a, b, and for each i, j,
4
4
0
0
ij
∫ du ∫ dv ρ ab (u, v) = 1 .
10
ij
ij
ρ ab
(u , v) is, therefore, a probability density on Ω. Let µ ab
be the corresponding probability
ij
measures. Each ρ ab
is nonzero only on the subspace Ej, and, on that subspace, takes the form
ij
ρ ab
(u , v) = σ ai (u ) τ bi (v) .
ij
is therefore, a sum of setting-dependent subspace product measures. In
For each pair i, j, µ ab
what follows, we denote the pair 〈i, j〉 by the single index m, which takes on 16 values.
m
That fact that each ρ ab
is, on the subspace picked out by m, a product of a function of u
and a and a function of v and b, might suggest that any correlations between u and v are due
solely to the shared time parameter m and that, in particular, the distribution of u is independent
of the distant setting b. This is not the case, because the subspace Ej picked out by m = 〈i, j〉 is
not a product space and has correlations between u and v built into it. In fact, each of the
m
measures µ ab
exhibits a rather extreme form of Parameter Dependence. Note that, for each
m
combination of m, a, b, the measure µ ab
assigns a nonzero probability to exactly one 1×1 square
in Ω, determined by the value of a and b. Furthermore, for given m, if one knows which 1×1
square has nonzero probability, one can ascertain the values of a and b. Thus, the dependence of
the marginal distribution of u on the value of b is such that knowing the values, on a given run of
the experiment, of u (which must be a value assigned nonzero probability) and m permits one to
ascertain both a and b. The same holds true of v. We therefore have Parameter Dependence—
for given m, changing the value of b affects the probability distributions for u, and changing the
value of a affects the probability distribution for v.
On the Hess-Philipp models, the corresponding Parameter Dependence is not quite as
extreme as that exhibited by our simple model. The values of m and u on a particular run of the
experiment, though they yield information about the setting of the distant apparatus, does not, on
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their model, permit one to ascertain precisely the value of the distant apparatus. The Parameter
Dependence is, however, present (this can be ascertained by inspection of equation [44] of Hess
and Philipp 2001a, or equation [27] of 2001c), and suffices for their model to reproduce the
statistical predictions of quantum mechanics.
Given the fact that, on our model, knowledge of m, u permits one to ascertain b, and
knowledge of m,v permits one to ascertain a, it would not be difficult, therefore, to cook up
functions Am(a,u), Bm(b,v) such that, for each m, the expectation value
∫ du ∫ dv ρ
m
ab
(u , v) Am (a, u ) Bm (b, v)
matches, for each of the four allowed combinations of the polarizers, the quantum mechanical
expectation value –a⋅b.
We will assume that associated with each measuring apparatus are synchronized timedependent parameters, and that m is a function of these, in such a way that, for any given run of
the experiment, the same value of m will be found at each station. We will also assume that, in
any period of time long enough to include many runs of the experiment, m ranges over all 16 of
its permitted values, in such a way that, for each run of the experiment, each value of m is
equally probable. We will therefore consider the probability measure,
µ ab =
1
∑ µ abm
16 m
which has density function
ρ ab (u , v) =
1
ρ abm (u , v) .
∑
16 m
It is easy to check that the measure µab is uniform over the square Ω. The fact that this is a
uniform measure, however, has no bearing on the locality of the model; it is a sum of measures;
each of which exhibits Parameter Dependence.
12
The Hess-Philipp model shares this feature with our simplified model. If the parameters
u, v, and m are observable, and if it is possible to use the value of m to adjust the setting of the a
measurement apparatus (and this is not excluded by anything they say), then signals can be sent
from one station to the other by adjusting the settings of the other, because, if one knows the
value of m for each run of the experiment (a value that is determined by correlated parameters at
each station), observation of the statistical distribution of u at station 1 yields information about
the setting b at station 2. If the parameters u, v, and m are not observable, this obscures the
nonlocality in the model but does not remove it; the dependence of the distribution of parameters
associated with one station on the setting of the other remains an inherent feature of the model.
The Hess-Philipp model achieves agreement with the quantum-mechanical predictions, not by
introducing parameters of a sort left out of consideration by Bell, but by using parameters of a
sort considered by Bell and violating the locality conditions assumed by Bell.
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Notes
1
I am grateful to Jim Brown for encouraging me to look at Hess and Philipp’s paper, and to
Richard Gill for helpful discussions. Since the completion of the draft of the version of this
paper originally submitted for presentation at PSA, two other criticisms of Hess and Philipp have
been put forward, that of Gill et al. (2002), and that of Mermin (2002). There is agreement
among these criticisms that Hess and Philipp have not shown that there is a flaw in the standard
proofs of Bell’s theorem, and that their model achieves its result via nonlocality. It is hoped that
the present paper is a useful supplement to the other criticisms and that, in particular, the
simplified model of section 4 helps to make it clearer what Hess and Philipp have done.
2
In a subsequent paper, Hess and Philipp remark that “Bell has included into his later proofs
(after publication of [Ballentine and Jarrett 1987]) setting-dependent parameters” (Hess and
Philipp 2002, 780). This remark, while true, is doubly misleading. First, it suggests that Bell did
not include such considerations prior to 1987. Second, it suggests that the cited paper is in some
way relevant to the consideration of such parameters. Ballentine and Jarrett’s excellent paper is,
however, about another topic entirely, namely, the distinction between “completeness” and
“locality” mentioned in section 2.
3
In a paper published subsequently to the papers presenting the model under discussion here,
Hess and Philipp remark, “Of course, to obey Einstein locality, [the station parameters] must be
station specific and can only be correlated by time-like correlations, i.e. by some relation to local
periodic processes” (Hess and Philipp 2002, 780). This is a clear statement of what their model
fails to do.
14
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